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We study the motion of a one-dimensional run-and-tumble particle with threediscrete internal states in the presence of a harmonic trap of stiffness $\mu.$The three internal states, corresponding to positive, negative and zerovelocities respectively, evolve following a jump process with rate $\gamma$. Wecompute the stationary position distribution exactly for arbitrary values of$\mu$ and $\gamma$ which turns out to have a finite support on the real line.We show that the distribution undergoes a shape-transition as$\beta=\gamma/\mu$ is changed. For $\beta<1,$ the distribution has adouble-concave shape and shows algebraic divergences with an exponent$(\beta-1)$ both at the origin and at the boundaries. For $\beta>1,$ theposition distribution becomes convex, vanishing at the boundaries and with asingle, finite, peak at the origin. We also show that for the special case$\beta=1,$ the distribution shows a logarithmic divergence near the originwhile saturating to a constant value at the boundaries.

Exact stationary state of a run-and-tumble particle with three internal states in a harmonic trap